Modifying additively manufactured part designs using topological analyses

ABSTRACT

Set differences between an as-designed and an as-manufactured model are computed. Topological deviations between the as-designed model and the as-manufactured model are determined based under-deposition and over-deposition features of the set differences. Based on the discrepancies, an input to a manufacturing instrument is changed to reduce topological differences between the as-manufactured model and the as-designed model.

RELATED PATENT DOCUMENTS

This application is a continuation of U.S. application Ser. No.16/235,229 filed on Dec. 28, 2018, which is incorporated herein byreference in its entirety.

STATEMENT REGARDING GOVERNMENT SUPPORT

This invention was made with government support under contract numberHR0011-17-2-0015 awarded by DARPA. The government has certain rights inthe invention.

SUMMARY

The present disclosure is directed to determining structural integrityof additively manufactured parts using topological analyses. In oneembodiment, set differences between an as-designed and anas-manufactured model are computed. Discrepancies between theas-designed model and the as-manufactured are determined basedunder-deposition (UD) and over-deposition (OD) features of the setdifferences. Based on the discrepancies, an input to a manufacturinginstrument is changed to reduce topological differences between theas-manufactured model and the as-designed model.

In another embodiment, motions of a manufacturing instrument arecomputed. As-manufactured models are computed from the motions bysweeping a minimum manufacturable neighborhood (MMN) over geometry of anas-designed model. Fields are computed over a configuration space(C-space) of the manufacturing instrument to determine overlaps of theMMN and the as-designed model. The motions of the manufacturinginstrument are parameterized by thresholding overlap measure fields. Theparameterized motions are used as an input to the manufacturinginstrument to create a modified replica of the as-designed model.

In another embodiment, UD and OD features are computed based on setdifferences of an as-designed and an as-manufactured model. Topologicalpersistence of the UD and OD features are computed in response tochanges in process parameters of a manufacturing instrument. Based onthe topological persistence, geometric and topological deviationsbetween the as-manufactured model from the as-designed model aredetermined. An input to a manufacturing instrument is changed based onthe deviations to reduce topological differences between theas-manufactured model and the as-designed model.

These and other features and aspects of various embodiments may beunderstood in view of the following detailed discussion and accompanyingdrawings.

BRIEF DESCRIPTION OF THE DRAWINGS

The discussion below makes reference to the following figures, whereinthe same reference number may be used to identify the similar/samecomponent in multiple figures. The drawings are not necessarily toscale.

FIG. 1 is a diagram showing various levels of overlap in an additiveprocess according to an example embodiment;

FIG. 2 is a diagram of a cross-correlation field for an as-designedshape and a minimum manufacturing neighborhood according to an exampleembodiment;

FIG. 3 is a diagram showing the effects of under and/or over depositionon shapes according to example embodiments;

FIG. 4 is a diagram showing a comparison between as-designed andas-manufactured shapes according to an example embodiment;

FIGS. 5 and 6 are diagrams showing under and over deposition regionsaccording to example embodiments;

FIG. 7 is a diagram illustrating a growing cell complex using inpersistent homology analysis according to an example embodiment;

FIG. 8 is a persistence barcode chart showing persistence of featuresshown in FIG. 7; and

FIG. 9 is a persistence diagram showing topological events according toan example embodiment;

FIG. 10 is diagram of a part used to illustrate analysis methodsaccording to an example embodiment;

FIG. 11 is a diagram showing cross-correlation fields and the resultingunder-deposition as-manufactured shapes for various additivemanufacturing resolutions according to an example embodiment;

FIG. 12 is a diagram of fields obtained as a summation of indicatorfunctions of as-manufactured shapes according to an example embodiment;

FIG. 13 is a plot visualizing the effects of two independent parameterson topology according to an example embodiment;

FIG. 14 is a diagram of field summations showing regions ofunpredictability indicated in FIG. 13;

FIG. 15 is a block diagram of a system according to an exampleembodiment; and

FIGS. 16-18 are flowcharts of methods according to example embodiments.

DETAILED DESCRIPTION

The present disclosure relates to additive manufacturing (AM). Additivemanufacturing has lifted many of the limitations associated withtraditional fabrication. Additive manufactured parts may include complexgeometric and topological structures and multi-material microstructuresto achieve improved performance such as high stiffness per weight, highsurface area per volume for heat transfer, and so on. Notwithstanding,the as-manufactured structures will differ from the as-designed in waysthat are difficult to characterize, quantify, and correct. Thesedeviations will depend on machine and process parameters. Note thathereinbelow the term “printing” or “3D printing” may be used to describeparticular additive manufacturing examples. However, this term is notintended to limit the embodiments to print-type processes only.

Previously, methods have been developed to identify, visualize, andcorrect non-manufacturable features in 3D printed parts. Although thesemethods provide visual and metrological information about the deviationsfrom intended design, they do not offer insight on the topologicalaspects. For many AM structures (e.g., lattices and foams) thetopological integrity of the structure can be important for the part toperform its functions. Even if the deviations are “small” from a metricpoint of view, they may lead to compromised function. For instance, ifthe beams in a lattice are slightly deformed due to the stair-steppingeffect of layered manufacturing, it may not matter as much as itsconnectivity. Similarly, addition or removal of tunnels and cavities canimpact performance (e.g., stress concentration under loads) orpost-processing (e.g., powder removal in DMLS).

The present disclosure relates to methods and systems configured tocharacterize the structural integrity of AM parts and quantify theas-manufactured deviations from the as-designed in terms of topologicalproperties. Quantifiers, such as Euler characteristic and Betti numbers,may be used to measure and certify such deviations. A family ofas-manufactured variants (rather than a single model) is obtained over arange of custom AM parameters such as manufacturing resolution andover-deposition allowance. Topological data analysis can be used toidentify topological deviations that persist across a relatively largerange of parameters. These methods and systems can be used to provide avisual and computational design tool to the user to reveal trade-offsbetween process parameters, as well as a diagnostics tool for designoptimization.

Practical limitations on the AM resolution and wall thickness introducegeometric and topological discrepancies between the as-designed targetand as-manufactured model. For example, the layer thickness in directmetal laser sintering (DMLS) is in the range 0.3-0.5 millimeters alongthe layer (XY-plan) and 20-30 microns along the build direction (Z-axis)at the highest resolution, with a minimum hole diameter of 0.90-1.15millimeters within a typical workspace of 250×250×325 millimeters-cubed.Attempting to fabricate designs that have smaller features may result indisconnected beams, filled holes/tunnels, or hard-to-predictcombinations. The as-manufactured resolution and wall thickness of the3D printer as shown in the part may eventually look and function quitedifferently from the as-designed part.

Methods may be used to model as-manufactured structures from a knowledgeof as-designed shape and AM parameters such as manufacturing resolutionand wall thickness. These methods may be used to generate AMprimitives/actions in hybrid manufacturing processes, which involveadditive combined and subtractive (e.g., machining) processes. The basicmodel of an as-manufactured shape is one that is obtained by sweeping aminimum manufacturable neighborhood (MMN) along an arbitrary motion thatis allowed by the machine's degrees of freedom (DOF). Many AMinstruments (such as 3D printers) operate by 3D translations of a movingpart (e.g., printer head) over the workpiece as it deposits a blob ofmaterial that is modeled by the MMN (e.g., an ellipsoid or a cylindroid)whose radii/height are determined by the printer resolution alongXYZ-directions. Unless the as-designed shape is not perfectly sweepableby the MMN along an allowable motion, the as-manufactured shape willdiffer. The motion need not be translational only (could be rotational,combination of rotations and translations, such as general rigid bodymotion, composition of multiple rigid body motions, etc.) The case oftranslations is merely an example, provided for purposes ofillustration.

One challenge is to find the “best” motion of the head whose sweep ofMMN results in a shape as close as possible to the as-designed target.The answer is not unique, as it depends on the notion of closeness.Closeness may be defined by criteria based on which the discrepanciesare measured or quantified.

This disclosure describes an approach to define and computeas-manufactured shapes. At every point in the 3D space inside theprinter workspace—which represents a translational motion of the printerhead—one obtains the overlap measure between the stationary as-designedshape and a MMN instance translated to the said point as the volume ofthe intersection region between them. The block diagram in FIG. 1 showsan MMN 100 overlapping a design geometry 102 according to an exampleembodiment. On the left-hand side of FIG. 1, the MMN 100 completelyoverlaps the geometry 102, and on the right-hand side the MMN 100partially overlaps the geometry. Generally, the overlap measure betweenthe as-designed part can change from zero to the total volume of the MMN100.

The measure of overlap can be thought of as a real-valued field definedover the configuration space (C-space) of relative motions (e.g.,translational, rotational, or combined) between the workpiece andmanufacturing instrument. For example, most commercial 3D printersoperate with translational DOFs by printing flat layers on top of eachother, in which case the overlap field can be viewed as a field over theprinter workspace in 3D. In such cases, the printability analysis can beperformed in at least two different ways. In a layer-by-layer analysis,the as-designed model is sliced along the build orientation into manylayers that are a constant distance apart (e.g., equal to printer'sknown layer thickness). For each 2D as-designed slice, a 2D field ofoverlap measures is constructed by using a 2D model of the MMN (e.g.,nozzle or laser beam cross-section). The measure is the surface area ofintersections between 2D shapes. The printability analysis may also usea full 3D analysis. In this type of analysis, a 3D field of overlapmeasures is obtained between the 3D as-designed model and a 3D model ofthe MMN (e.g., a blob of material that is representative of a depositionunit). In this case, the layer thickness and build orientation areencoded into the shape of the MMN. The measure is the volume ofintersections between 3D shapes.

The layer -by-layer analysis approximates the full 3D analysis when thelayer thickness is much smaller than the characteristic size of the MMNcross-section (e.g., a thin disk) so that volumetric overlap can beapproximated by the area overlap times the thickness. This is the casein many processes. For instance, there is one order of magnitudedifference in XY- and Z-resolutions in DMLS. It has been shown that theoverlap measure field can be computed cumulatively for all translationalmotions by a cross-correlation of indicator (e.g., characteristic)functions of the as-designed shape and MMN in 3D—or theirslices/cross-sections in 2D.

The indicator function of a shape is a field in its underlying spacethat maps every point in that space to a binary value; namely, 0 if thepoint is outside and 1 if the point is inside the shape. It can beviewed as an implicit model of the shape that characterizes pointmembership queries about that shape. Cross-correlation is a standardoperation on integrable fields that is computed as a convolution of thefirst field—indicator function of an as-designed shape—with a reflectionof the second field—indicator function of the MMN. The convolution, inturn, can be computed rapidly using fast Fourier transforms (FFT) forwhich highly efficient parallel implementations exist on both CPU andGPU architectures.

In general, the AM instrument can move according to arbitrary DOF (e.g.,both translational and rotational) in which case the overlap measure isdefined over a higher-dimensional C-space. For example, in addition to3D printing with flat layers, robotic 3D printers with higher-DOF havebeen prototyped—e.g., to enable support-free 3D printing on adaptivelyreoriented platforms. Moreover, “multi-tasking” machines for hybrid(combined additive and subtractive) manufacturing are becomingincreasingly popular. These machines use the computer numericallycontrolled (CNC) motion system—originally developed for high-axismachining—for AM as well, enabling deposition of material on non-flatsurfaces (e.g., around a cylindrical shaft) using rotational or combinedrotational and translational motions.

The methods described herein are not restricted to translationalmotions. For example, for arbitrary subgroups of the group of rigidmotions (e.g., combined translations and rotations), the convolutionformulation of the overlap measure generalizes to group-theoretic notionof convolutions. The FFT-based implementation is more complicated toreconcile. However, partial speed-ups can be obtained by using theunique properties of the rigid motion group—e.g., by computing adifferent overlap measure field for sparsely sampled orientations.

The different superlevel-sets of the cross-correlation (the overlapmeasure) field give a family of totally-ordered sets (ordered by setinclusion) of configurations in the C-space (relative translationsand/or rotations). The members of the family can be distinguished by asingle parameter; namely, the overlap measure that changes between zeroand total measure of the MMN—or an overlap measure ratio between 0 and 1after normalizing it by the maximum value that can be assumed constantfor most AM processes. At every choice of the parameter (e.g., if onechooses a ratio of 35%), all configurations of the MMN that lead to atleast that overlap measure ratio are included in the as-manufacturedshape, e.g., all displacements of the MMN that make at least 35% of itsvolume overlapped by the as-designed part. For translational motion, theminimum and maximum superlevel sets corresponding to 0% and 100% overlapmeasure ratios are the morphological dilation and erosion and can becomputed using Minkowski sum and difference. For higher-DOF motions withrotations, these notions generalize to the C-space obstacle and itscomplement (free space) and can be computed using Minkowski products andquotients as well as lifting and projection maps between Euclidean 3Dspace and configuration space.

In FIG. 2, a diagram shows an example 2D cross-correlation field 208 foran as-designed shape 200 and a MMN 202 according to an exampleembodiment. The cross-correlation field gives the overlap measure—inthis case the area of the intersection regions 210-214 shown for fiveconfigurations of the MMN 202. Different superlevel-sets of thecross-correlation field correspond to a family of configuration sets(e.g., motions)—in this case, 2D translations collected into thepointsets 204-206. Sweeping the MMN 202 along each of these motionsleads to a one-parametric family of printable (e.g., MMN-sweepable)shapes that are morphologically close to the target as-designed shape200.

For each one of these configuration sets 204-206, the as-manufacturedshape is obtained by sweeping the MMN along the set, which ischaracterized as another dilation and can be computed as a Minkowskisum/product depending on the motion DOF. The one-parametric family ofas-manufactured alternatives form a totally ordered set in the 3D space(in terms of set containment) bounded by the two extremes.

At a first extreme, the sweep of MMN along configurations with slightlyunder 100% overlap leads to a strict under-deposition/under-fill policy.In this case, the as-manufactured shape is the maximal depositableregion (MDR)—with the specified DOF and MMN of the 3D printer—that iscompletely contained inside the as-designed part. This guarantees thatall containment constraints for which the part was originally designed(e.g., remaining inside some envelope while moving in assembly) willremain satisfied in spite of as-manufactured deviations. In fact, theconstraints are satisfied with minimal possible compromise in shape dueto AM limitations.

At a second extreme, the sweep of MMN along configurations with slightlyover 0% overlap leads to a strict over-deposition/over-fill policy. Inthis case, the as-manufactured shape fully contains the as-designed withan extra offset. This is useful as a global allowance to generate anear-net shape that fully contains the as-designed shape and can belater machined down to get closer to the as-designed shape.

There is a spectrum of possibilities for the as-manufactured shape whenthe allowance is relaxed by choosing a value of the overlap measureratio between 0 and 1. Every decrease in the overlap measure ratio growsthe as-manufactured shape by a non-uniform offset that depends on thelocal geometry of both the as-designed shape and MMN. For translationalmotions, the under-fill shape is a morphological opening (dilation oferosion) while the over-fill shape is a double-offset (dilation ofdilation). The continuous family of as-manufactured shapes in between,parameterized by the overlap measure ratio, will have small geometricdeviations from the as-designed part as the MMN gets smaller in radius.However, they can have dramatically different topological properties.

The changes in the machine DOF, MMN size/shape, and the criteria onoverlap measures can introduce topological discrepancies to theas-manufactured shape(s) from the as-designed shape, regardless of howsmall the geometric deviations are. A method and system can beconfigured to characterize the differences in basic topologicalproperties of an arbitrary as-designed shape and an as-manufacturedshape—including but not necessarily restricted to as-manufactured modelscomputed using the methods explained above.

One concept utilized in these systems and methods is the notion of Eulercharacteristic (EC) of the shapes, which compactly characterizes some ofthe main topological properties of any shape. For example, if the shapeis discretized with a cellular complex, the EC is obtained as EC=V−E+Fwhere V, E, and F are the total number of vertices, edges, and faces inthe complex, respectively. Another relationship that is useful for thesepurposes is EC=B0−B1+B2 where B0, B1, and B2 are the Betti numbers (BN)of the shape. In 3D, the BNs correspond to the number of connectedcomponents, tunnels (i.e., through-holes), and voids/cavities,respectively.

In FIG. 3, a diagram shows differences in BNs between as-designed andas-manufactured shapes according to an example embodiment. The BNs (onlyB0 & B1 in 2D) are shown in FIG. 3 for a few simple 2D examples ofas-designed vs. as-manufactured shapes with under- and over-depositionpolicies. The EC and BNs are global topological properties. Therefore,comparing them between the as-designed and as-manufactured shapes doesnot provide much information on local topological discrepancies and whatfeatures are responsible for them. For example, beams thinner than theMMN diameter could break when using an under-deposition policy. However,the structure may remain globally connected through other links across,as shown by respective as-designed and as-manufactured shapes 300, 302.In this example, under deposition may lead to broken beams but that mayor may not affect total number of connected components (B0) or tunnels(B1), which correspond to holes for the 2D case. In this case two holesmerge with the external empty space (B1 reduces by 2). The EC increasesby 2 as a result.

Tunnels or cavities may be covered if they are smaller than the MMNdiameter when using an over-deposition policy, but they may be part of alarger hole or tunnel that remains topologically intact. As seen byrespective as-designed and as-manufactured shapes 304 and 306, overdeposition may lead to covered holes but that may or may not affecttotal number of connected components (B0) or holes (B1). In this casetwo disconnected components join and a new hole appears, leading to adecrease in B0 and an increase in B1. The EC does not change as aresult.

In more complicated scenarios, multiple holes can merge in strange wayswhile new holes appear, keeping the total number of holes the same. Asseen by respective as-designed and as-manufactured shapes 308 and 310,it is possible in both under deposition or under deposition (or combinedpolicies) to see no change in BNs, hence no change in EC either. In thiscase, a hole splits into two and another hole gets covered at the sametime.

In general, the global EC and/or BNs may remain the same and countingtheir global changes is insufficient to detect local topologicaldiscrepancies. Even when they do change, their values provide no insightinto what features might have caused those changes and how to fix themby either changing the design or the manufacturing process parameters(e.g., MMN size/shape). Methods described herein can provide a moredetailed and localized account of topological discrepancies between theas-designed and as-manufactured shapes. The methods can provide aprecise report on which features or their boundaries with the rest ofthe part caused which defect(s) and how they can be fixed by using adifferent policy that is specialized for that feature—e.g.,over-deposition for thin beams and under-deposition for covered holes.

One of the useful properties of EC is its additivity. If a shape isdecomposed into a union of several other (possibly intersecting) shapes,its EC can be computed by an alternating sum expressed as: sum of theECs of all components, minus EC of pairwise intersections, plus EC oftriplewise intersections, and so on. When two shapes are beingcompared—namely, the as-designed (D) and as-manufactured (M) shapes—thedifference between their ECs is quantified in terms of ECs of theunder-deposition (UD) and over-deposition (OD) regions and theirboundaries with D and M. The global topological deviation as a functionof the local topological properties of these regions can be expressed asin Equation (1) below.

EC[M]−EC[D]=(EC[OD region]−EC[OD's cut boundary])−(EC[UD region]−EC[UD'scut boundary])   (1)

The OD is defined as the region in space that is obtained as theregularized set difference of D from M, which is roughly the set of 3Dpoints that are inside M but outside D (the over-deposited material).This set sticks out of the original design and is connected to the restof M along pieces of the boundary of D. These pieces are collectivelycalled the OD's “cut boundary,” an example of which is shown as dashedlines on shape 306 in FIG. 3.

The UD is defined as the region in space that is obtained as theregularized set difference of M from D, which is roughly the set of 3Dpoints that are inside D but outside M (the under-deposited material).This set sticks inside the original design and is connected to the restof D along pieces of the boundary of M. These pieces are collectivelycalled the UD's “cut boundary,” an example of which is shown as dashedlines on shape 302 in FIG. 3.

As seen in FIG. 4, an example is provided of arbitrary changes from theas-designed shape 400 to as-manufactured 402 according to an exampleembodiment. This can be modeled using any method of choice, not justthresholding overlap measures (e.g., values of cross-correlationfields). As seen in FIGS. 5 and 6, for arbitrary changes in the globaltopology of the shape, the deviations are identified in terms ofcontributions of local OD & UD features. Formula (1) above can beexpanded further in terms of the ECs of the different connectedcomponents of each of OD and UD. Noting that the different connectedcomponents of any shape are (by definition) the disjoint connectedshapes (shapes that do not intersect) whose union is the original shape(in this case the overall UD or OD under consideration), the EC of theshape can be expressed as the sum of ECs of connected components due toEC's additivity.

Consider the simpler case in which each of these connected componentsare homeomorphic to a ball, a volume having no tunnels or cavities. Inthat case EC=B0−B1+B2 reduces to EC=1 (since B1=0 and B2=0). Thus, theEC of the volumetric portion of the feature is equal to unity for eachconnected component of OD and UD. However, it may or may not be unityfor the cut boundaries, which will determine whether that feature iscontributing a nonzero value to the overall difference between the totalnumber of connected components of as-designed and as-manufactured parts.Only if the cut boundary is a single simply-connected patch for a givenvolumetric component 500, the contribution of the feature (composed ofthe volumetric component paired with the cut boundary) will be zero tothe overall EC. If the cut boundary is, for example, multiplesimply-connected patches 600, or a hollow closed surface (topologicalsphere) 602, or multiply-connected complex surfaces, the contributionwill be nonzero. In other words, the decomposition enables one toquantify the contribution of each local change to the total globalchange. Even if any of the global BNs is intact due to the kinds ofnontrivial exchanges between local elements discussed earlier (e.g., asin shapes 308 and 310 in FIG. 3), the local changes themselves will berevealed and quantified. The local changes are quantified by analyzingthe different connected components of the symmetric difference of D andM (the OD and UD pieces) and their cut boundaries. In a nutshell, thetwo terms in Equation (1) are expanded as shown in Equations (2) and (3)below, where CB stands for cut boundary.

(EC[OD region]−EC[OD's CB])=sum of (EC[OD piece]−EC[OD piece's share ofCB]) for all OD pieces   (2)

(EC[UD region]−EC[UD's CB])=sum of (EC[UD piece]−EC[UD piece's share ofCB]) for all UD pieces   (3)

Substituting the above identities into the earlier formula leads to thefollowing conclusion: The total change in EC of the shape due tomanufacturing imperfections can be obtained as the sum of totalcontributions of OD pieces minus the sum of total contributions of UDpieces. Each piece's contribution can be computed independently from theothers, which enables the computation by perfectly parallel processing.

For the general case where each connected component is not simplyconnected and contractible (e.g., B1 and B2 are nonzero as well hence ECis not the same as B0), each connected component can be broken down intosmaller pieces until each smaller piece has EC=1 for the volumetricinterior and the negative contributions of the cut boundary are computedfor each piece. However, the smaller pieces will not be disjoint, asthey will share boundaries with each other, unlike the case withdisjoint (bigger) connected components. In this case, one requires moreterms in the above two formulae to add/subtract the EC of the sharedboundaries in the form of an alternating sum. To solve this, eachconnected component of OD and UD is efficiently decomposed into simplyconnected pieces using various standard techniques. The boundariesbetween these pieces (pairwise, triplewise, and so on) can be obtainedfrom the shape's Reebs graph, from which the additional terms in theabove formulae are systematically included. The alternative approach isto use any standard method to compute all BNs per original OD and UDpiece and their cut boundaries.

Once the contributions of different pieces to the overall EC isdetermined, the method reasons about the different modes ofmanufacturing failure due to local topological discrepancies. Moreover,it provides feedback on locally changing the deposition policy locallyor the as-designed structure itself to alleviate the problems. Based onthe type of difference regions between D and M (OD versus UD) and thesign and value of their contributions to BNs (e.g., adding/removing oneor more connected components, tunnels, or cavities) different classes oftopological defects and potential remedies are identified.

For example, if a UD feature contributes a positive amount to the totalnumber of connected components (B0) and/or a negative amount to thetotal number of tunnels (B1), it suggests a broken beam (e.g., shape 302in FIG. 3). One remedy is to apply an OD policy in the vicinity of thatparticular feature—e.g., using the cross-correlation methods explainedearlier, but this time applying it locally to the UD feature rather thanthe entire design.

If an OD feature contributes a negative amount to the total number ofconnected components (B0) and/or a positive amount to the total numberof tunnels (B1), it suggests a covered tunnel (see shape 306 in FIG. 3).One remedy is to apply a UD policy in the vicinity of that particularfeature—e.g., using the cross-correlation methods explained earlier, butthis time applying it locally to the OD feature rather than the entiredesign.

Even when none of the BNs change (in which case, neither does EC as aresult), this method can still detect nonzero contributions of thedifferent features to the total zero change. For example, it will revealthe case where a hole gets covered while two other holes get merged,leading to a net change of zero in B1 (see shape 310 in FIG. 3). This isbecause multiple OD and/or UD connected components will appear in theset difference between D and M and their precise role in the changes inBNs. In particular for this example, an increase in B1 due to onefeature and a decrease in B1 due to another feature that cancels it outin the net sum are detected regardless of the net sum being zero. Forthe purpose of illustration, 2D examples are shown and described above,however the theory generalizes to 3D and the method works for bothmanufacturability analysis of 2D slices and the whole 3D shape alike.Note that the machine DOF do not affect the method's applicability as itworks for arbitrarily complex D and M shapes.

The above analysis detects topological discrepancies between theas-designed and as-manufactured shapes and identifies their spatialdistribution in terms of local contributions from UD and OD features. Italso provides useful information to make local changes to the design orMMN to eliminate those contributions. However, this analysis cannotdetect how important each of these contributions are relative to oneanother. It considers a single as-manufactured outcome; hence provideslittle insight on how persistent they are across the spectrum of allpossible as-manufactured alternatives that one can obtain by changingthe AM process specs or deposition policies—e.g., by changing theshape/size of MMN, threshold on overlap measure ratio, etc. If oneconsiders changes in a one -parametric family of as-manufactured shapesas a continuous evolution along a time-like axis, a temporal analysiscan be used in addition to the spatial analysis given in previoussection to provide insight into the level of impact that local changescan have on the as-manufactured shape.

Persistent homology is a powerful computational tool for topologicaldata analysis, first introduced for continuous topologicalsimplification through one-parametric filtration of a growing cellcomplex. The present method can use persistent homology to capturepersistent topological features of as-manufactured shapes as they arevaried across the different parameters such as MMN size or overlapmeasure ratio. Given a number of data points in an arbitrary-dimensionalmetric space, one can think of a number of growing balls centered ateach point (of the same radius) which characterize an influence regionaround the point with respect to the chosen metric.

An example of how growing feature size affects topology is seen in FIG.7. As the radius r is increased, the balls grow and start intersectingeach other increasingly more. One can think of an abstract simplicialcomplex—called the Vietoris-Rips (or simply Rips) complex—thatcharacterizes these intersections. For example, if two balls intersect,a 1-cell (an edge) is drawn that connects the vertices assigned to theircenters. Hence if three balls intersect pairwise but not triplewise, onesees an empty triangle between them. If they also intersect triplewise,a 2-cell (filled triangle) can be drawn between them. This process iscalled topological “filtration” and the growing radius is sometimescalled the proximity/filtration parameter.

As the shape of the union of balls and its representative Rips complexevolve, its topological properties change. A topological “event” isdescribed as the birth/death of a topological feature such as a mergingtwo connected components into one, filling of a hole/tunnel or cavity,and so on. If the filtration is thought of as an evolution over time,the BNs can be plotted over time as well as bars that indicate the timeinterval between birth and death of features (one bar per (birth, death)pair of events). The so-obtained “persistence barcode” is often dividedinto pieces, one for each homology group corresponding to BNs. Anexample of a persistence barcode is shown in FIG. 8, which plots thebirth/death of components (B0) and holes (B1) in FIG. 7 as the radiusincreases.

The persistence barcode is a topological signature of the data withregards to its distribution in the metric space. A “persistence diagram”can also be used to visualize the results in which the birth and deathtimes are plotted against each other in a 2D plot, an example of whichis shown in FIG. 9. The farther from the intercept line 900 the (birth,death) point is, the more distant in time those events are, and the morepersistent (hence fundamental) that feature is perceived to be. Thuspoint 902 in FIG. 9 is associated with the feature that has the longestpersistence. The fundamental topological features of data are thosewhich persist over longer periods of filtration. On the other hand,features that are not persistent (e.g., point 904) can be due tonoise/errors in computation, and are less likely to be of significance.The length of bars in persistence barcode can similarly provide anindication of the fundamental features.

Persistent homology has since been applied to various different types ofcell complexes and filtration methods. For example, a superlevel-set ofa function can be viewed as a filtration based on which a cubic cellcomplex that represents a voxelization of the set can be studied for itsevolving topological properties/events.

As applied to AM, this technique may be applied using two differentfiltrations. First, for a fixed deposition policy (e.g., constantoverlap measure ratio in our own cross-correlation method), a filtrationis provided by changing the size of the MMN. This can be done byapplying uniform scaling on a fixed-shape MMN, but any other shapeparameterization that corresponds to a realistic family of growing MMNsfor one or more 3D printers can be used. Second, for a fixed MMNshape/size (e.g., fixed 3D printer with fixed specs), a filtration isprovided by changing the deposition policy. This can be done by usingthe overlap measure ratio (between 0 and 1) as the filtration parameter,but any other filtering that produces a total ordering can be used.

There are more recent methods developed for persistent homology ofmulti-variate filtrations which can be applied to analyze thesimultaneous changes in both MMN shape/size and deposition policy. Thefollowing discussion will focus on a single-parametric filtration thatchanges one parameter while keeping the other fixed, starting from MMNsize. This method will be illustrated on 2D as-designed slices andas-manufactured slices obtained as superlevel-sets of the 2Dcross-correlation field obtained using a 2D cross-section of a thin-diskMMN. However, the exact same approach can be applied to the full 3Danalysis.

In FIG. 10, a diagram shows an example shape 1000 used to demonstratepersistent homology methods according to example embodiments. The shapehas relatively complex geometry but a simple topology for illustrationpurposes (homeomorphic to a disk; EC=B0−B1=1). The method applies toarbitrary topology but this example shape 1000 is chosen to simplify theillustration and not for limitation. The shape 1000 has severalinteresting geometric features such as sharp corners and thin flanks.Although it appears to have holes, they are not topological holesbecause they are connected to external space through a thin channel.This results in a simply-connected shape with BNs B0=1 & B1=0.

Using a strict under-deposition policy with a simple disk-shaped MMNwith varying radius, a family of as-manufactured slices are computed asthe morphological openings of the as-designed slice with each disk. Thisleads to “filleting” the corners of the as-designed slice that aresharper than the MMN radius and eliminating the features that arethinner than the MMN diameters. An example applied to the shape 1000from FIG. 10 is shown in FIG. 11. The overlap measure(cross-correlation) fields 1100 are shown for different, disk-shapedMMNs 1100. The min/max level-sets are also shown as curves, e.g., curves1006, 1008 respectively. The max superlevel-sets 1008 represent themaximal set of translational configurations of the MMN that are fullycontained inside the as-designed slice (strict under-deposition).Sweeping the MMN along this set gives the white areas in shapes 1104,whereas the hatched regions are non-manufacturable.

For the smallest MMN (effectively a point) the non-manufacturable regionis empty (not shown) and the as-manufactured shape is identical to theas-designed shape. As the radius is increased from this ideal case, theas-manufactured shape shrinks (from right to left) until it completelydisappears when the MMN diameter is larger than the thickest feature inthe as designed shape (left). Along this evolution, topological featuresare born or die, and are captured by this method.

The larger the MMN radius that is picked, the smaller theas-manufactured slice will be as it is constrained by the chosen policyto be completely contained inside the as-designed slice. Moreover, thefamily of one-parametric filleted slices are totally ordered by setcontainment meaning that every increase in fillet radius leads to aslice that is fully contained inside an earlier slice with a smallerfillet radius. This is true for arbitrary shapes of MMN (provable byset-theoretic arguments), if the MMN increases in size in a way that itcontains an earlier MMN, the resulting morphological opening iscontained inside the one obtained with the earlier MMN.

As the as-manufactured shape gets smaller and starts deviating from theas-designed due to imposed disconnections that happen along thinfeatures within which the MMN does not fit anymore. At certain“critical” values of the MMN size (disk radius/diameter in this example)topological events occur such as the birth of two or more connectedcomponents from one, death of two or more holes by merging into one,etc. The methods described herein provide a detailed account of theseevents by performing persistent homology.

Whenever such a totally ordered family is given, one can assemble theminto a field by arithmetic summation of their indicator functions. Eachsuperlevel-set of this field corresponds to one as-manufactured shape,hence the persistent topological features can be identified by applyingpersistent homology to this field using a superlevel-set filtration. Thefield can be represented discretely on a uniform grid of pixels (2Dimage) or voxels (3D image) and calling a standard persistent homologycode written for cubic cell complexes. This is the simplest and arguablymost efficient/general implementation since cross-correlations andmorphological operations are extremely fast on image representationsusing FFT-based convolutions as explained above. If a differentdiscretization of the fields is preferred for any particular use-case,the analysis can be done on the corresponding cell complex in a similarfashion.

Now consider the second filtration (based on overlap measure ratio). Fora fixed MMN, the cross-correlation readily provides a field, computedusing a single FFT-based convolution. Superlevel-sets of this field areextracted for different overlap measures. For each set, theas-manufactured slice is obtained as a sweep (dilation) with the MMN.The indicator functions of the resulting family of shapes are once againsummed up to obtain the field on which the same persistent homologyalgorithm can be applied using a superlevel-set filtration as seen inFIG. 12. The only thing that changes from the previous procedure is theinput field. Each field in FIG. 12 represents a totally ordered familyof as-manufactured slices for the as-designed slice 1000 in FIG. 10using a constant radius of the MMN (e.g., fixed AM process resolution,measured in image pixels) but a range of overlap measure ratios (between70% and 100% in this example). Darker points correspond to higheroverlap measure ratios. The darkest black regions indicate strict-underdeposition, and the brightest grey indicates a 30% allowance forover-deposition.

If the analysis is performed for filtration based on one parameter(e.g., overlap measure ratio) while keeping the other one fixed (e.g.,MMN size) and the algorithm is repeated for different values of thelatter, a number of persistence barcodes/diagrams are obtained thatprovide a detailed account of expected topological behavior for a rangeof different AM processes with varied resolution and allowance specs.

To simplify the visualization of the enormous amount of information,first consider the global topological properties of the as-manufacturedshape and how it evolves with different combinations of the twoindependent parameters. One of the BNs can be plotted (for example, thenumber of connected components B0) as a function of the two parametersplaced on the two axis of a 2D plot. An example of this is shown in FIG.13.

In FIG. 13, plots show the number of connected components (B0) for theas-manufactured slices plotted (left) using different allowance(abscissa) and MMN radii (ordinate) for the example as-designed slice1000 in FIG. 10 (B0=1). On the two plots on the right, each horizontalline 1300 and vertical line 1302 represents trade-offs between geometricand topological losses. For a fixed resolution, the intersection of thehorizontal line 1300 corresponding to fixed MMN radius with the boundaryof the large region 1304 in the plot gives the minimum requiredallowance for the printed slice to keep its as-designed connectivityproperties (B0).

For a fixed allowance, the intersection of the horizontal line 1304corresponding to fixed overlap measure ratio with the boundary of theregion 1304 in the plot gives the minimum required resolution for theprinted slice to keep its as-designed connectivity properties (B0). Theintersection of these lines with the other regions each indicate the“critical” values of the printing parameters at which some as-designedtopological property (in this case, connectivity) is about to becompromised.

This plot provides a useful visualization of the trade-offs that areinevitable when using AM to approximately build a given as-designedshape. For example, using a low-resolution 3D printer (larger MMN) willinevitably result in geometric deviations from the as-designed shape.Increasing the allowance for over -deposition (smaller overlap measureratio threshold) could further introduce geometric deviations that onemay or may not be able to remove later—e.g., using machining. However,the plot shows that the connectivity can be recovered by adding theallowance if one can tolerate the additional geometric inaccuracies. Inother words, when topological integrity is more important—which is oftenthe case, e.g., in infill lattices that have to remain connected forstructural strength or porous for heat convection—our method provides atool to analyze how much sacrifice is needed in geometric accuracy tomaintain topological properties. This is illustrated by the horizontallines 1300 in FIG. 13.

If there are strict restrictions on the geometric accuracy—e.g., due tocontainment constraints in assembly or limited accessibility formachining the near-net shape—the allowance has to remain small. The plotshows what it takes in terms of minimum AM process resolution if theconnectivity is to remain intact. In other words, it reveals the minimumMMN size for a given maximum allowance for which the topologicalintegrity can be preserved. This is illustrated by the horizontal lines1300 in FIG. 13.

Note that each of the boundaries between different shaded regions inFIG. 13 can be viewed as a Pareto frontier. They reveal the trade-offsbetween resolution (MMN size) and allowance (overlap measure ratio) fora tolerable limit/bound on the deviation of a topological property(e.g., some BN). For example, the first curve 1306 is the Pareto frontfor zero tolerance on the difference in the BN of connected componentsbetween as-designed and as-manufactured shapes. The second curve 1308 isa Pareto front for the case in which the most that can be tolerated isto change the BN by 1, which will add/remove one connected component,tunnel, or cavity, depending on which BN is plotted. For every fixedfront, the plot gives the designer visual feedback on what the optionsare with respect to AM process resolution and allowance, so he/she canchoose what to do depending on what is available on the manufacturingshop floor, tolerance specs, etc.

Another important information revealed by the plot is the singularities(e.g., regions 1310) that represent the (resolution, allowance)combinations that should be avoided. These combinations lead to unstablecomputational results in modeling the topological properties of theas-manufactured shape. If the computations are reasonable approximationsto the reality of AM—which is the assumption for everything else to makesense—then this means that the actual output of the AM process may beunpredictable in these cases.

For example, as shown in the example fields shown in FIG. 14, when theMMN diameter is very close to the thickness of a noncircular feature,the overlap measure cannot be reliably computed especially due torasterization errors. If the AM instrument (e.g., a digital 3D printer)also follows these computational algorithms to determine its motionalong each slice, e.g., the set of motions identified as superlevel-setsof cross-correlation, the physical outcome will be unpredictable aswell.

Similar 2D plots can be generated for other BNs or EC to providefeedback to the designer about the variations of global topologicalproperties with respect to AM parameters and the inherent trade-offsbetween geometric accuracy and topological integrity. The same analysiscan be performed on the contributions of local features to the globalvariations as discussed in an earlier section. The latter is not onlyfavorable due to focusing attention on the precise spatial locations ofpotential topological defects, it is computationally more practicalbecause local features can be represented by smaller cubic complexes andthe persistent homology algorithm terminates in a much shorter period oftime. Hence this method can provide real-time feedback to the designeras they vary the parameters and see its effects on the BN plots for eachpotentially problematic feature. The designer can accordingly makecorrections locally or globally and try again.

In summary, the present disclosure describes methods and systems used toquantify and qualify structural integrity of additively manufacturedparts by the aid of a digital computer. An example system utilizingthese methods is shown in FIG. 15. The system 1500 includes one or moreprocessors 1501, memory 1502 (e.g., random access memory) and persistentstorage 1504 (e.g., disk drive, flash memory). These components 1501,1502, 1504 are coupled by data transfer lines (e.g., memory busses,input/output busses) and are configured to perform functions indicatedby functional module 1506. The processor 1501 may include anycombination of central processing units and graphical processing units.

The module 1504 receives input data (e.g., CAD geometry) that describesan as-designed shape 1506 of arbitrary complexity. The module 1504 alsoutilizes parameters 1508 that may describe an arbitrary representationand specifications of an AM process in terms of its MMN and depositionpolicy (e.g., allowance for over-deposition). Based on this data, 1506,1508, the module 1504 obtains the indicator functions of the inputrepresentations of the as-designed target 1506 and MMN, which is a 3Dfield that assigns a binary number to every point in the 3D space (1/0for points that are inside/outside the shape, respectively). Aconvenient 3D field representation is given by voxelization—akin toimages with binary pixels but in 3D—but any other field representationcan be used. Or a 3D volume can be decomposed into 2D and processed asdescribed above.

The following analysis can be performed either on full 3D model or onits 2D slices for a specific build orientation and slicing parameters(e.g., layer thickness). If the latter is chosen by the user or deemedsuitable by the algorithm, the indicator functions of the slices ofas-designed shape and MMN slice are obtained from the indicatorfunctions of their 3D models. The cross-correlations of the twoindicator functions are computed, which is a convolution of the firstindicator function (as-designed shape or one of its slices) with thereflection of the second indicator function (MMN or its cross-section).This produces a real-valued field in 3D space that can be interpreted asthe overlap measure of the MMN with the as-designed shape (the totalvolume/area of their intersection in 3D/2D) if the MMN is moved todifferent positions in the 3D/2D space that contains the as-designedshape/slice.

Next, desired superlevel-set(s) 1510 of the obtained overlap measure(cross-correlation) field corresponding to the desired deposition policyis obtained. This can range from strict under-deposition—where theas-manufactured shape/slice is required to be fully contained inside theas-designed shape/slice—to conservative over-deposition where theas-designed shape/slice is thickened everywhere. They correspond to themax/min superlevel-sets of the cross-correlation field. There are afamily of mixed under- and over-deposition possibilities in betweenthese two extremes that are obtained as superlevel-sets parameterized bypartial overlap measure ratios. Each set represents a potentialcandidate for as-manufactured shape/slice that is close to theas-designed shape/slice but has both geometric and (possibly)topological deviations from it due to AM limitations.

The output 1510 from this part of the software can be used as input tothe next steps described below. However, the rest of the algorithm worksfor arbitrary as-manufactured shapes/slices. Hence, if the user choosesto call an external module to use different algorithms based ondifferent criteria to generate the as-manufactured shape/slice (or afamily of them) they can do so, import the result, and the rest of themethod will work without any changes.

Given the as-designed shape/slice and one possible as-manufacturedshape/slice that is computed in any arbitrary way, the global and localtopological discrepancies can be quantified between the shapes/slices aswell as the contribution of different features to the deviations andpossible remedies to recover the lost topological properties. Forexample, the following can be computed: the OD and UD regions (thedifference regions between the as-designed and as-manufacturedshapes/slices); the intersection region between the as-designed andas-manufactured, and boundaries that the intersection regions share withthe as-designed and as-manufactured shapes/slices. Once again, voxelrepresentations may be used for the ability to rapidly compute their ECand BN.

The OD and UD regions can be decomposed into their connected componentssuch that “cut boundaries” are identified. The cut boundaries may bedefined as the boundaries the over- and under-deposition regions sharewith the as-designed and as-manufactured shapes/slices, respectively.The contributions of each connected component and its cut boundary tothe total topological deviation between as-designed and as-manufacturedshapes/slices can be computed in terms of an additive topologicalproperty (e.g., EC) as well as possibly other properties that add up toobtain the said topological property (e.g., BN, number of cells in acell complex, etc.). This can be done in parallel for every connectedcomponent.

The connected components can be composed into smaller components thathave simpler topology (e.g., no tunnels/cavities) and theircontributions to the original component characterized in terms ofalternating sums. This can be done using any standard method (e.g.,Reebs graphs). Components that contribute nonzero values to the globaltopological property(s) are identified. Topological defects caused fromdifferent combinations of the sign and values of nonzero contributionsfor OD and UD regions can be classified. For each type of problem, alocal remedy can be prescribed—e.g., change the policy locally toeliminate the problem or give feedback (plus suggested design changes)to the designer for the local feature.

In some embodiments, the persistence of topological properties can bequantified across a range of AM parameters such as printer resolutionand allowance for over-deposition. For example, persistent homology tocan be used to compute a persistence barcode/diagram of the topologicalevents that occur such as the birth/death of topological features alongcontinuous change of the AM parameters. This process (called“filtration”) identifies the parameter values at which theas-manufactured shape/slice experiences a change in its topologicalproperties that (possibly) further deviates it from the as-designedshape/slice. The filtration parameters include, but are not limited to,the size of the MMN and allowance for over-deposition (e.g., overlapmeasure ratio).

The above global procedure can be performed on local regions such as theOD and UD components that individually contribute to the overalltopological changes. The trade-offs between geometric accuracy andtopological integrity can be visualized by plotting the global and/orlocal topological properties as a function of the different filtrationparameters. The system 1500 also includes a user interface 1512 that canbe used to both receive user inputs (e.g., varying parameters 1516 thatwould theoretically be input to a manufacturing instrument 1514 such asan AM instrument) and present a visualization of the effect of the userinputs on the topological changes to the as-manufactured part.

One example implementation of this method is to plot BNs as a functionof MMN size and allowance for over-deposition (e.g., overlap measureratio) for the global as-manufactured family of shapes/slices or thelocal OD and UD features (see, e.g., FIG. 13). This plot shows thetrade-off between resolution and allowance, or viewed differently,between geometric inaccuracies tolerated by over-deposition andtopological disintegration. For example, if it is possible to recover anas-designed topological characteristic (e.g., connectivity or porosityin a local region) by sacrificing geometric accuracy, this plot willshow the minimum required sacrifice. Similarly, it will provide minimumresolution required to maintain topology when there are constraints onmaximal allowance, and vice versa. This enables defining theminimal/maximal AM parameter values that give different levels ofguarantees to limit/bound topological deviations.

For example, this could be used to inform the designer that in order tokeep the change in the number of connected components (similarly fortunnels or cavities) less than or equal to a certain amount (includingzero: no change allowed) what is the minimum resolution (for fixedallowance) or minimum allowance (for fixed resolution) required. Moregenerally, it provides a Pareto frontier of trade-offs between theseparameters for every required bound on the topological deviations(locally or globally). A collection of parameters may also be identifiedthat are likely to produce computationally unreliable results due tonoise and/or representation approximations (e.g., rasterization error),hence could produce unpredictable output after sending theserepresentations to an AM instrument. The designer can be provided withfeedback to avoid these combinations of parameters.

In FIG. 16, a flowchart shows a method according to an exampleembodiment. The method involves computing 1600 set differences betweenan as-designed and an as-manufactured model. One or more UD and ODfeatures are computed 1601, e.g., by decomposing set differences betweenthe UD and OD features into disjoint pieces. Discrepancies between theas-designed model and the as-manufactured model are determined based onthe set differences, e.g., by looking at properties of the disjointpieces. For example, local topological properties (e.g., EC, BN) of theUD and OD features can be computed and combined to obtain contributionsto the global properties. The discrepancies can be quantified based onthe local topological properties. Based on the discrepancies, an inputto a manufacturing instrument is changed to reduce topologicaldifferences between the as-manufactured model and the as-designed model.

In FIG. 17, a flowchart shows a method according to another exampleembodiment. The method involves computing 1700 motions of amanufacturing instrument (e.g., an AM instrument). As-manufacturedmodels are computed 1701 from the motions by sweeping a MMN over thegeometry of an as-designed model. Fields are computed 1702 over aconfiguration space of the manufacturing instrument to determineoverlaps of the MMN and the as-designed model (e.g., viacross-correlations or convolutions of defining functions of the MMN andthe as-designed shapes). The motions of the manufacturing instrument areparameterized 1703 by thresholding overlap measure fields. Theparameterized motions are used 1704 as an input to the manufacturinginstrument to create a modified replica of the as-designed model.

In FIG. 18, a flowchart shows a method according to another exampleembodiment. The method involves computing 1800 UD and OD features basedon set differences of an as-designed and an as-manufactured model.Topological persistence of the UD and OD features are computed 1801 inresponse to changes in process parameters of a manufacturing instrument(e.g., an AM instrument). Based on the topological persistence,geometric and topological deviations between the as-manufactured modelfrom the as-designed model are determined 1802. An input to themanufacturing instrument is changed 1803 based on the deviations toreduce topological differences between the as-manufactured model and theas-designed model.

The various embodiments described above may be implemented usingcircuitry, firmware, and/or software modules that interact to provideparticular results. One of skill in the arts can readily implement suchdescribed functionality, either at a modular level or as a whole, usingknowledge generally known in the art. For example, the flowcharts andcontrol diagrams illustrated herein may be used to createcomputer-readable instructions/code for execution by a processor. Suchinstructions may be stored on a non-transitory computer-readable mediumand transferred to the processor for execution as is known in the art.The structures and procedures shown above are only a representativeexample of embodiments that can be used to provide the functionsdescribed hereinabove.

Unless otherwise indicated, all numbers expressing feature sizes,amounts, and physical properties used in the specification and claimsare to be understood as being modified in all instances by the term“about.” Accordingly, unless indicated to the contrary, the numericalparameters set forth in the foregoing specification and attached claimsare approximations that can vary depending upon the desired propertiessought to be obtained by those skilled in the art utilizing theteachings disclosed herein. The use of numerical ranges by endpointsincludes all numbers within that range (e.g. 1 to 5 includes 1, 1.5, 2,2.75, 3, 3.80, 4, and 5) and any range within that range.

The foregoing description of the example embodiments has been presentedfor the purposes of illustration and description. It is not intended tobe exhaustive or to limit the embodiments to the precise form disclosed.Many modifications and variations are possible in light of the aboveteaching. Any or all features of the disclosed embodiments can beapplied individually or in any combination are not meant to be limiting,but purely illustrative. It is intended that the scope of the inventionbe limited not with this detailed description, but rather determined bythe claims appended hereto.

1. A method, comprising: determining a family of as-manufactured shapesbased on an as-designed shape of a part, each member of the familycomprising different overlap measure ratios between one and zerorespectively corresponding to strict over-deposition and strictunder-deposition policies of an additive manufacturing instrument;determining under deposition (UD) regions and over deposition (OD)regions based on intersection regions between the as-designed shape andthe as-manufactured shapes; decomposing the UD regions and the ODregions into connected components; computing contributions of theconnected component to a total topological deviation between theas-designed shaped and as-manufactured shapes in terms of an additivetopological property; composing the connected components into smallercomponents that have simpler topologies; characterizing thecontributions of the smaller components to the to the respectiveconnected component in terms of alternating sums of the additivetopological property to identify a subset of the connected componentsthat contribute nonzero values to a global topological property; foreach of the subset of connected components, providing localized remediesto reduce the contributions of the connected components to the globaltopological property; and changing an input to the additivemanufacturing instrument to include the localized remedies when buildingthe as-manufactured shape.
 2. The method of claim 1, wherein decomposingthe UD regions and the OD regions into the connected components definescut boundaries of the connected components, the intersection regionssharing the cut boundaries with the as-designed shape and theas-manufactured shapes, and wherein the cut boundaries are usedcomputing contributions of the connected components to the totaltopological deviation.
 3. The method of 1, wherein determining thefamily of as-manufactured shapes comprises cross-correlations orconvolutions of defining functions of a minimum manufacturableneighborhood (MMN) and the as-designed shape.
 4. The method of claim 1,wherein determining the family of as-manufactured shapes comprises:determining a first indicator function based on the as-designed shape;determining a second indicator function based on a minimummanufacturable neighborhood (MMN) of the additive manufacturinginstrument; determining a cross-correlation of the first indicatorfunction with the second indicator function, the cross-correlationproviding a real-valued field corresponding to an overlap measure of theMMN with the as-designed shape; and selecting a superlevel set of thereal-valued field that defines the family of as-manufactured shapes. 5.The method of claim 4, wherein the first and second indicator functionsare fields in the respective underlying spaces that map every point inthe underlying spaces to 0 if points are outside the underlying spacesand to 1 if the points are inside the underlying spaces.
 6. The methodof claim 1, wherein composing the connected components into smallercomponents that have simpler topologies connected components into: ODfeatures that are within the as-manufactured shapes and outside theas-designed shape; OD cut boundaries that connect the OD features to aboundary of the as-manufactured shapes; UD features that are within theas-designed shape and outside the as-manufactured shapes; and UD cutboundaries that connect the UD features to the boundary of theas-manufactured shapes.
 7. The method of claim 6, wherein the additivetopological properties comprise Euler characteristics (EC) of: the ODfeatures; the OD cut boundaries; the UD features; and the UD cutboundaries.
 8. The method of claim 1, wherein the localized remedycomprises one or both of applying an OD policy to a UD feature andapplying a UD policy to an OD feature, wherein the OD policy and UDpolicy are applied only to the vicinity of the respective UD and ODfeature, and not to the entire as-manufactured shape.
 9. A systemcomprising a processor configured via instructions to perform the methodof claim
 1. 10. A method comprising: determining a family ofas-manufactured shapes based on an as-designed shape of a part, eachmember of the family comprising different overlap measure ratios betweenone and zero respectively corresponding to strict over-deposition andstrict under-deposition policies of an additive manufacturinginstrument; determining localized topological discrepancies for each ofthe as manufactured shapes as compared to the as-designed shape, thelocalized topological discrepancies contributing nonzero values to aglobal topological property; perform topological persistence filteringover a continuous change of process parameters of the additivemanufacturing instrument; based the topological persistence filtering,determining a combination of the process parameters that result atolerable bound on the localized topological discrepancies; changing aninput to the additive manufacturing instrument to utilize thecombination of the process parameters.
 11. The method of claim 10,wherein the topological persistence filtering comprises changing thesize of a minimum manufacturable neighborhood of the additivemanufacturing instrument for a given deposition policy.
 12. The methodof claim 10, wherein the topological persistence filtering compriseschanging deposition policies of the additive manufacturing instrumentfor a given minimum manufacturable neighborhood.
 13. The method of 4,wherein determining the family of as-manufactured shapes comprisescross-correlations or convolutions of defining functions of a minimummanufacturable neighborhood (MMN) and the as-designed shape.
 14. Themethod of claim 13, wherein determining the family of as-manufacturedshapes comprises: determining a first indicator function based on theas-designed shape; determining a second indicator function based on aminimum manufacturable neighborhood (MMN) of the additive manufacturinginstrument; determining a cross-correlation of the first indicatorfunction with the second indicator function, the cross-correlationproviding a real-valued field corresponding to an overlap measure of theMMN with the as-designed shape; and selecting a superlevel set of thereal-valued field that defines the family of as-manufactured shapes. 15.The method of claim 15, wherein the first and second indicator functionsare fields in the respective underlying spaces that map every point inthe underlying spaces to 0 if points are outside the underlying spacesand to 1 if the points are inside the underlying spaces.
 16. A systemcomprising a processor configured via instructions to perform the methodof claim 10.